Add Email Signature

I have a mail which is not having the signature appended. Why not?

This probably means that the plugin sending the mail is using its own mechanism to send it, instead of calling wp_mail as it should. Request the plugin author to instead use WordPress's official mechanism.

The signature is only being added to the plain text of an email, and not inside an HTML mail

That's right. If you have some code that's capable of delving inside the HTML to do this correctly, then I'll be pleased to use it. But I couldn't think of a way.